# Matching of Orbital Integrals on \(GL(4)\) and \(GSp(2)\)

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*Yuval Z. Flicker*

The trace formula is the most powerful tool currently
available to establish liftings of automorphic forms, as predicted by
Langlands principle of functionality. The geometric part of the trace
formula consists of orbital integrals, and the lifting is based on the
fundamental lemma. The latter is an identity of the relevant orbital
integrals for the unit elements of the Hecke algebras.

This volume concerns a proof of the fundamental lemma in the
classically most interesting case of Siegel modular forms, namely the
symplectic group \(Sp(2)\). These orbital integrals are
compared with those on \(GL(4)\), twisted by the transpose
inverse involution. The technique of proof is elementary. Compact
elements are decomposed into their absolutely semi-simple and
topologically unipotent parts also in the twisted case; a double coset
decomposition of the form \(H\backslash G/K\)—where H is a subgroup
containing the centralizer—plays a key role.

#### Table of Contents

# Table of Contents

## Matching of Orbital Integrals on $GL(4)$ and $GSp(2)$

- Contents vii8 free
- Abstract viii9 free
- Introduction 110 free
- Part I. Preparations 615 free
- Part II. Main comparison 3948
- A. Strategy 3948
- B. Twisted orbital integrals of type (I) 3948
- C. Orbital integrals of type (I) 4453
- D. Comparison in stable case (I), E/F unramified 4756
- E. Comparison in stable case (I), E/F ramified 5160
- F. Endoscopy for H = GS[sub(p)](2), type (I) 5362
- Unstable twisted case. Twisted endoscopic group of type I.F.2 5867
- Twisted endoscopic group of type I.F.3, E/F unramified 5968
- G. Twisted orbital integrals of type (II) 6170
- H. Orbital integrals of type (II) 6574
- I. Comparison in case (II), E/E[sub(3)] ramified (e = 2) 6978
- Unstable twisted case. Twisted endoscopic group of type I.F.2 7281
- J. Comparison in case (II), E/E[sub(3)] unramified (e = 1) 7382
- Unstable twisted case. Twisted endoscopic group of type I.F.3 7786
- K. Endoscopy for GS[sub(p)](2), type (II) 7988
- L. Comparison in case (III) 8291
- Unstable twisted case. Twisted endoscopic group of type I.F.2 8695
- M. Comparison in case (IV) 8897
- Unstable twisted case. Twisted endoscopic group of type I.F.2 97106

- Part III. Semi simple reduction 100109
- References 111120